On the application of two symmetric Gauss Legendre quadrature rules for composite numerical integration over a triangular surface

摘要

This paper first presents a Gauss Legendre quadrature rule for the evaluation of I=∫∫Tf(x,y)dxdy, where f(x,y) is an analytic function in x, y and T is the standard triangular surface: {(x,y)|0⩽x,y⩽1,x+y⩽1} in the two space (x,y). We transform this integral into an equivalent integral ∫∫Sf(x(ξ,η),y(ξ,η))∂(x,y)∂(ξ,η)dξdη where S is the 2-square in (ξ, η) space: {(ξ,η)|-1⩽ξ,η⩽1}. We then apply the one-dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient Quadrature rules with new weight coefficients and new sampling points. Then a second Gauss Legendre quadrature rule of composite type is obtained. This rule is derived by discretising T into three new triangles TiC(i=1,2,3) of equal size which are obtained by joining centroid of T, C=(1/3,1/3) to the three vertices of T. By use of affine transformations defined over each TiC and the linearity property of integrals leads to the result:I=∑i=13∫∫TiCf(x,y)dxdy=13∫∫TG(X,Y)dXdY,where G(X,Y)=∑i=1n×nf(xiC(X,Y),yiC(X,Y)) and x=xiC(X,Y) and y=yiC(X,Y) refer to affine transformations which map each TiC into T the standard triangular surface. We then write ∫∫TG(X,Y)dXdY=∫∫SG(X(ξ,η),Y(ξ,η))∂(X,Y)∂(ξ,η)dξdη and a composite rule of integration is thus obtained. We next propose the discretisation of the standard triangular surface T into n2 right isosceles triangular surfaces Ti (i=1(1)n2) each of which has an area equal to 1/(2n2) units. We have again shown that the use of affine transformation over each Ti and the use of linearity property of integrals lead to the result:∫∫Tf(x,y)dxdy=∑i=1n×n∫∫Tif(x,y)dxdy=1n2∫∫TH(X,Y)dXdY,where H(X,Y)=∑i=1n×nf(xi(X,Y),yi(X,Y)) and x=xi(X,Y), y=yi(X,Y) refer to affine transformations which map each Ti in (x, y) space into T a standard triangular surface T in the (x, y) space. We can now apply the two rules earlier derived to the integral ∫∫TH(X,Y)dXdY, this amounts to application of composite numerical integration of T into n2 and 3n2 triangles of equal sizes respectively. We can now apply the rules, which are derived earlier to the evaluation of the integral, ∫∫Tf(x,y)dxdy and each of these procedures converges to the exact value of the integral ∫∫Tf(x,y)dxdy for sufficiently large value of n and the convergence is much faster for higher order rules. We have demonstrated this aspect by applying the above composite integration method to some typical integrals.